In hyperbolic geometry, the order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron.
The dihedral angle of a Euclidean regular dodecahedron is ~116.6ð, so no more than three of them can fit around an edge in Euclidean 3-space. In hyperbolic space, however, the dihedral angle is smaller than it is in Euclidean space, and depends on the size of the figure; the smallest possible dihedral angle is 60ð, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72ð.
There are four regular compact honeycombs in 3D hyperbolic space:
There is another honeycomb in hyperbolic 3-space called the order-4 dodecahedral honeycomb, {5,3,4}, which has only four dodecahedra per edge. These honeycombs are also related to the 120-cell which can be considered as a honeycomb in positively curved space (the surface of a 4-dimensional sphere), with three dodecahedra on each edge, {5,3,3}. Lastly the dodecahedral ditope, {5,3,2} exists on a 3-sphere, with 2 hemispherical cells.
There are nine uniform honeycombs in the [5,3,5] Coxeter group family, including this regular form. Also the bitruncated form, t<sub>1,2</sub>{5,3,5}, , of this honeycomb has all truncated icosahedron cells.
The SeifertâÂÂWeber space is a compact manifold that can be formed as a quotient space of the order-5 dodecahedral honeycomb.
This honeycomb is a part of a sequence of polychora and honeycombs with icosahedron vertex figures:
This honeycomb is a part of a sequence of regular polytopes and honeycombs with dodecahedral cells:
The rectified order-5 dodecahedral honeycomb, , has alternating icosahedron and icosidodecahedron cells, with a pentagonal prism vertex figure.
There are four rectified compact regular honeycombs:
The truncated order-5 dodecahedral honeycomb, , has icosahedron and truncated dodecahedron cells, with a pentagonal pyramid vertex figure.
The bitruncated order-5 dodecahedral honeycomb, , has truncated icosahedron cells, with a tetragonal disphenoid vertex figure.
The cantellated order-5 dodecahedral honeycomb, , has rhombicosidodecahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.
The cantitruncated order-5 dodecahedral honeycomb, , has truncated icosidodecahedron, truncated icosahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure.
The runcinated order-5 dodecahedral honeycomb, , has dodecahedron and pentagonal prism cells, with a triangular antiprism vertex figure.
The runcitruncated order-5 dodecahedral honeycomb, , has truncated dodecahedron, rhombicosidodecahedron, pentagonal prism, and decagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.
The runcicantellated order-5 dodecahedral honeycomb is equivalent to the runcitruncated order-5 dodecahedral honeycomb.
The omnitruncated order-5 dodecahedral honeycomb, , has truncated icosidodecahedron and decagonal prism cells, with a phyllic disphenoid vertex figure.